Given two $q$-ary codes $C_1$ and $C_2$, the relative hull of $C_1$ with respect to $C_2$ is the intersection $C_1\cap C_2^\perp$. We prove that when $q>2$, the relative hull dimension can be repeatedly reduced by one, down to a certain bound, by replacing either of the two codes with an equivalent one. The reduction of the relative hull dimension applies to hulls taken with respect to the $e$-Galois inner product, which has as special cases both the Euclidean and Hermitian inner products. We give conditions under which the relative hull dimension can be increased by one via equivalent codes when $q>2$. We study some consequences of the relative hull properties on entanglement-assisted quantum error-correcting codes and prove the existence of new entanglement-assisted quantum error-correcting maximum distance separable codes, meaning those whose parameters satisfy the quantum Singleton bound.

翻译：给定两个 $q$ 进制码 $C_1$ 和 $C_2$，则 $C_1$ 相对于 $C_2$ 的相对壳层定义为交集 $C_1\cap C_2^\perp$。我们证明：当 $q>2$ 时，通过将两个码之一替换为等价码，相对壳层维数可逐步减少一，直至达到某一界。该相对壳层维数的缩减适用于以 $e$-Galois 内积定义的壳层，而欧几里得内积与埃尔米特内积均为其特例。我们给出了当 $q>2$ 时通过等价码将相对壳层维数增加一的条件。本文研究了相对壳层性质对纠缠辅助量子纠错码的影响，并证明了存在新的纠缠辅助量子纠错最大距离可分码，即参数满足量子 Singleton 界的码。